How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial #P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3# have a real zero between the numbers 0 and 1?

1 Answer
Aug 29, 2015

Answer:

You can't.

Explanation:

You can use synthetic division (and the Remainder Theorem) to find

#P(0) = 3# (although using division seems like the long way to do this)

and #P(1) = 3#

If we one of these positive and the other negative, then, since #P(x)# is continuous, we could use the Intermediate Value Theorem to conclude that there is a zero in #(0,1)#.

The Intermediate Value Theorem can never be used to conclude that there is not a zero in an interval.

We could conclude that #P(x)# does not have exactly one zero between #0# and #1#. (It must have an even number of zeros in in #(0,1)#.)

To help explain why, here is the graph of a different polynomial function:

graph{ x^4 + 2x^3 + 2x^2 - 5x +1 [-3.067, 3.09, -1.17, 1.91]}

Call this polynomial function #Q(x)#

Notice that #Q(0) = 1# and also #Q(1) = 1# but his function does have a zero in #(0,1)#. (It has two of them.)